Question regarding the Many-Worlds interpretation

Let's prepare and experiment with individual photons moving along the z-axis and polarization in the xy-plane such that a detector registers
- polarization along the x-axis with 90% and
- polarization along the y-axis with 10%

According to the MWI for each registered photon there's a branching such that in every branch one experimental result (either x- or y-polarization) is realized. Let's repeat the experiment N times with N individual photons which results in 2N branches in total.

Now my problem is that according to the branching the observers expect a result "y-polarization" with 50% probability, whereas according to the experimental setup they expect a result "y-polarization" only with 10% probability. Of course we know that in practice the second calculation is correct, such that I as an observer will always live in a branch where the result set "xyxxyxxxx..." agrees with the 90%-10% probabilities.

How does the MWI resolve this contradiction?
How does the MWI forces me to find myself in one of the most probable branches?

The short answer is that it doesn't. That's one of the fundamental problems of Everett style interpretations: The count of observed events does not emerge from the dynamics itself but has to be added in form of an additional postulate. You can either directly postulate an observed frequency proportional to the squared amplitude magnitude, or you can postulate something more subtle and then derive the probabilities. This is what Deutsch/Wallace based on decision theory do, or Zurek based on stability assumptions and the ad-hoc existence of a probability measure.

I think that the arguments to derive the Born rule from MWI are deeply flawed. Not even logically, but they implicitly admit that emergence is not enough to deal with the measurement problem and undermine the very basis of the idea. This makes the concept of many worlds much harder to accept, because it doesn't take you the whole way and in the end you still have to make additional assumptions just like in any other proposed solution of the measurement problem.

Of course we know that in practice the second calculation is correct, such that I as an observer will always live in a branch where the result set "xyxxyxxxx..." agrees with the 90%-10% probabilities.

The MWI doesn't agree. You will not always live in such a branch. There's no mechanism which makes a certain branch which agrees with the probabilities more "real" than a branch where you always get y. It is only much more likely to end up in such a branch.

But you are right that the emergence of the probabilities in the first place is a problem. As Jazzdude has pointed out, there isn't a universally accepted derivation of the Born rule.

I think that the arguments to derive the Born rule from MWI are deeply flawed. Not even logically, but they implicitly admit that emergence is not enough to deal with the measurement problem and undermine the very basis of the idea. This makes the concept of many worlds much harder to accept, because it doesn't take you the whole way and in the end you still have to make additional assumptions just like in any other proposed solution of the measurement problem.

In thinking about the Born rule and MWI, I have wondered what it would even mean to derive the Born rule for MWI. In some "possible worlds", relative frequencies for outcomes of experiments will be well-described by the Born rule, and in other possible worlds, they won't. The best you can do is to come up with a measure on the possible worlds such that the Born rule is valid for "most" worlds (according to that measure), but the only real measure we have is the Born rule itself. So that's kind of circular, but I don't know how it could be otherwise.

This is a philosophical problem that I think is common to all "ensemble" theories of probability--the theory isn't really falsifiable. Just about any outcome for any experiment is consistent with the ensemble theory.

Now my problem is that according to the branching the observers expect a result "y-polarization" with 50% probability

This is a common fallacy. Where did the number 50% come from? Who said the branches have to be equally probable? It's like saying I have a 50/50 chance of winning a lottery: either I win or I don't.

Consider a simple example: you send a random photon through a polarizing beam splitter. There are 2 branches, in one branch (branch 1) the photon goes left, in another (branch 2) it goes right. They both exist simultaneously but you have a 50/50 chance to be in one branch or the other. Now consider another beam splitter at 45 degrees in the right output path. Now we have 3 branches (1, 2a, 2b), branch 1 remains as it was, while branch 2 splits into 2a and 2b when the photon hits second polarizer. Clearly if you have a 50% chance to get into branch 2 and then another 50/50 chance to get into either 2a or 2b, then you have an overall 25% chance to end up in branch 2a. In other words, each of the branches 2a and 2b is twice "thinner" compared to branch 1.

There are derivations of Born rule form symmetry considerations. Some people claim the problem is solved, while other people say it is based on circular argument. Ensuing discussion usually descends into the abyss of philosophy:)

@All: I don't see how additional assumptions or postulates can resolve this contradiction. We know from QM and experiment that most observers will observe "x", but counting branches and observers only 50% will observe "x". So the problem is that in the majority of branches sequences "xyxxyyxyxxyyxyx" with "50% x - 50% y" are observed. Therefore most branches are incompatible with our observations.

I don't see how additional assumptions or postulates can resolve this contradiction.

I fully agree. The only "natural" way to come up with a probability would involve branch counting. The counter argument of the Everettians contains two main arguments: 1) That is an ad-hoc assumption about the attribution of "reality" to the branches. 2) Branch counting is impossible in general, as the number of branches depends on a number of things and is highly subjective. So a probability metric depending on branch counting could not deliver objective probabilities.

To me 2) is also saying that the whole world equals branch idea is not applicable. Either we can talk about different discrete realities or we cannot. In the latter case MWI has a much more fundamental problem than the emergence of the Born rule.

Their answer is however that we simply have to postulate a probability measure for the branches that also gives the branches a weight that equals out all possible issues with branch identification, so fixing both problems at the same time. The Born rule would do that, if you believe them, so it must be the correct way to measure subjective probabilities of finding yourself inside a branch.

Here's a little thought experiment that illustrates my issues with the Born rule in MWI: Consider we have two copies of a universe, both containing the same history of branches but with different squared magnitudes. In one universe the history is very likely, in the other the history is very unlikely as dictated by the Born rule. However the final states of both histories in the final branch are identical, only their amplitude differs. Consequently, the relative evolution and information contained in both branches is identical. Now if the Born rule would really follow from the amplitudes only, then in one branch the observer would say "Now that outcome I observe was very likely" whereas in the other branch he would say "That was rather unlikely!". But they cannot say different things, because their branches are dynamically identical and not correlated to the history of the amplitudes. This is a direct consequence of the linearity of the evolution.

You can clearly see, I'm not an Everettian. But I would be if it delivered what it promised: A theory of structural emergence from just unitary global evolution.

Now my problem is that according to the branching the observers expect a result "y-polarization" with 50% probability

I'm asking again where did the number 50% come from. Specifically starting from branch counting how did you arrive at the probability of being in particular branch. It appears that at some point in doing so you assume equal probability for the branches. This assumption is completely unfounded.

In reality, there is always a lot of stuff going on. Consider a photomultiplier. A photon hitting a detector triggers an avalanche of electrons. Each time a collision happens, there is a "branching" (MWI) / "wavefunction collapse" (Copenhagen), whichever you prefer. By the time the signal the appears at the output, the world has split / wavefunctions collapsed many times over. So instead of 2 separate branches (photon X/photon Y), we in fact have a gazillion of branches, differing between each other in minute details, like the number and positions of all the electrons knocked out at each stage. Roughly 0.9 gazillion of branches (roughly) will have the photon in state X and another 0.1 gazillion in state Y. If you are in the game of counting branches you should count all these.

So the problem is that in the majority of branches sequences "xyxxyyxyxxyyxyx" with "50% x - 50% y" are observed. Therefore most branches are incompatible with our observations.

These outcomes are possible but have a very low probability of occurring. Which just means that the majority of branches have very small measure (which is another way of saying exactly the same thing).

Staff: Mentor

I don't see how additional assumptions or postulates can resolve this contradiction. We know from QM and experiment that most observers will observe "x", but counting branches and observers only 50% will observe "x". So the problem is that in the majority of branches sequences "xyxxyyxyxxyyxyx" with "50% x - 50% y" are observed. Therefore most branches are incompatible with our observations.

The branches aren't incompatible with observation, as the outcome in any single branch is a physically possible outcome. The problem is that the ratios of numbers of branches of a given x:y ratio doesn't lead to a prediction of the probabilities that matches observation; and from that I can only conclude that either those ratios have no physical significance or (as Delta Kilo suggests) we aren't counting them right.

Staff: Mentor

In reality, there is always a lot of stuff going on. Consider a photomultiplier. A photon hitting a detector triggers an avalanche of electrons. Each time a collision happens, there is a "branching" (MWI) / "wavefunction collapse" (Copenhagen), whichever you prefer. By the time the signal the appears at the output, the world has split / wavefunctions collapsed many times over. So instead of 2 separate branches (photon X/photon Y), we in fact have a gazillion of branches, differing between each other in minute details, like the number and positions of all the electrons knocked out at each stage. Roughly 0.9 gazillion of branches (roughly) will have the photon in state X and another 0.1 gazillion in state Y. If you are in the game of counting branches you should count all these.

That's one way of explaining the observed macroscopic probabiities, given that we only get to investigate a single path through the tree. However, I find it less than completely satisfying (full disclosure: it is unlikely that anything will ever induce me to say anything nice about MWI) as in principle I ought to be able to separate the first interaction from the cascade by an almost arbitrary distance. Say I prepare a particle in the spin-up state then pass it through a Stern-Gerlach device oriented at 45 degrees that will route it off towards either of two very distant galaxies. Because the SG device is at 45 degrees not 90, the probability that the particle will be detected at one galaxy a million years later is different than the probability of its detection at the other - but how is this difference encoded in the branching that MWI says happens at the SG device? It's somewhat tempting to look for internal state in the particle itself, such that there are more "goes left" branches than "goes right" branches from the initial interaction with the SG apparatus.

Jazzdude: would you not solve that by interpreting worlds as divergent rather than splitted? like alastairwilson.org does?

No, that doesn't really change anything. At some point of the divergence process you must be able to separate the worlds by dynamic independence and you must be able to count them. Otherwise the whole concept is flawed.

The problem is that most observers exist in branches with arbitary low probability calculated using the "90% - 10% - rule". In the above mentioned example the branch "xx" has a probability of 81% but 3/4 of all observers exist in branches with only 19% in total. Changing probabilities (preparing different superpositions) does affect the QM probabilities but does not affect the branching.

So repeating and correcting myself the problem is that the ratio of the number of branches is incompatible with our observations.

Your idea that counting branches means that one should count all these branches and that the majority of branches have very small measure is correct, of course, but I hope you agree that MWI should then provide a means to define branch counting, a probability measure, a derivation of the QM probabilities, or at least a means to assign compatible measures (instead of simply claiming that the branches with small measure have small QM probability w/o being able to define or derive these statements). It seems that you want to hide these problems by introducing rather complex branch structures and ending up with a result of roughly 0.9 gazillion of branches having the photon in state X and another 0.1 gazillion in state Y. I would agree to this result, iff you are able to present a derivation of this result.

Let's prepare and experiment with individual photons moving along the z-axis and polarization in the xy-plane such that a detector registers
- polarization along the x-axis with 90% and
- polarization along the y-axis with 10%

So in this experiment 1000 photons hit 1000 detectors. X-axis polarization is detected by 900 detectors. Y-axis polarization is detected by 100 detectors.

When a detector detects one of the thousand photons, for some reason it becomes unable to detect the 999 other photons. Probably this has something to do with "branching".

Of course in reality the number of photons and detectors is much larger than 1000.

The above is either the standard many worlds interpretation or my wrong idea about the standard many worlds interpretation.

According to the MWI for each registered photon there's a branching such that in every branch one experimental result (either x- or y-polarization) is realized. Let's repeat the experiment N times with N individual photons which results in 2N branches in total.

A quantum wave can be thought as superposition of many identical quantum waves.
A quantum wave of a detector can be thought as superposition of many identical quantum waves of a detector.
A detector can be thought as superposition of many identical detectors.

In the experiment the thicker of the two resulting branches contains 10 times more detectors than the other branch.

So instead of 2 separate branches (photon X/photon Y), we in fact have a gazillion of branches, differing between each other in minute details, like the number and positions of all the electrons knocked out at each stage. Roughly 0.9 gazillion of branches (roughly) will have the photon in state X and another 0.1 gazillion in state Y. If you are in the game of counting branches you should count all these.

That is quite a strong assertion. And it is fundamentally incompatible with the linearity of the state evolution in quantum theory. The weights of the initial superposition cannot have influence on the number of states in either category. Do the math and see for yourself.

I hope you agree that MWI should then provide a means to define branch counting, a probability measure, a derivation of the QM probabilities, or at least a means to assign compatible measures.

Compatible measures are assigned according to the Born rule, just like in any other interpretation. As for the rest, yes, it would be very nice to have, but then it would make MW a theory instead of interpretation.

It seems that you want to hide these problems by introducing rather complex branch structures ...

I didn't mean to suggest any particular branch counting scheme that works. On the contrary I was just trying to show that branch counting is meaningless unless a) you can say something about relative measures of the branches (eg from symmetry considerations) and b) there is an unambiguous way to count the branches.